×

On the dynamics of a higher-order rational recursive sequence. (English) Zbl 1235.39001

Summary: We investigate the global convergence result, boundedness, and periodicity of solutions of the recursive sequence \[ x_{n+1}=ax_{n}+\dfrac{bx_{n-l}+cx_{n-k}}{dx_{n-l}+ex_{n-k}},\;\;\;n=0,1,\dots, \] where the parameters \(a,b,c,d\;\)and\(\;e\;\)are positive real numbers and the initial conditions \(x_{-k},x_{-k+1},\dots,x_{-l},x_{-l+1},\dots,x_{-1}\) and \(x_{0}\;\)are positive real numbers.

MSC:

39A10 Additive difference equations
39A22 Growth, boundedness, comparison of solutions to difference equations
39A30 Stability theory for difference equations
PDFBibTeX XMLCite
Full Text: Euclid

References:

[1] R. P. Agarwal, Difference Equations and Inequalities, 1\(^{st}\) edition. Marcel Dekker, New York. 1992, 2\(^{nd}\) edition. 2000.
[2] R. P. Agarwal and E. M. Elsayed, Periodicity and stability of solutions of higher order rational difference equation. Advanced Studies in Contemporary Mathematics. 17 (2) (2008), pp 181-201. · Zbl 1169.39001
[3] M. Aloqeili, Dynamics of a rational difference equation. Appl. Math. Comp. 176 (2) (2006), pp 768-774. · Zbl 1100.39002 · doi:10.1016/j.amc.2005.10.024
[4] E. Camouzis, A. Gilbert, M. Heissan and G. Ladas, On the Boundedness Character of the System \(x_{n+1}=\dfrac{\alpha _{1}+\gamma _{1}y_{n}}{x_{n}},\;y_{n+1}=\dfrac{\alpha _{2}+\beta _{2}x_{n}+\gamma _{2}y_{n}}{A_{2}+x_{n}+y_{n}}\). Commun. Math. Anal. 7 (2) (2009), pp 41-50. · Zbl 1166.39003
[5] C. Cinar, On the positive solutions of the difference equation \( x_{n+1}=\dfrac{x_{n-1}}{-1+ax_{n}x_{n-1}}.\;\)Appl. Math. Comp. 158 (3) (2004), pp 793-797. · Zbl 1069.39022 · doi:10.1016/j.amc.2003.08.139
[6] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, Global attractivity and periodic character of a fractional difference equation of order three. Yokohama Math. J. 53 (2007), pp 89-100. · Zbl 1138.39006
[7] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, On the difference equation \(\;x_{n+1}=ax_{n}-\dfrac{bx_{n}}{cx_{n}-dx_{n-1}}.\) Advances in Difference Equations. Volume 2006 , Article ID 82579, pp 1-10. · Zbl 1139.39304 · doi:10.1155/ADE/2006/82579
[8] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, On the difference equations \(x_{n+1}=\dfrac{\alpha x_{n-k}}{\beta +\gamma \prod_{i=0}^{k}x_{n-i}}.\) J. Conc. Appl. Math. 5 (2) (2007), pp 101-113. · Zbl 1128.39003
[9] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, Global behavior of the solutions of difference equation, Advances in Difference Equations , · Zbl 1271.39008
[10] E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, On the solutions of difference equations of order four, Rocky Mountain J. Math. , · Zbl 1355.39003
[11] E. M. Elabbasy and E. M. Elsayed, On the global attractivity of difference equation of higher order. Carpathian J. Math. 24 (2) (2008), pp 45-53. · Zbl 1174.39304
[12] E. M. Elabbasy and E. M. Elsayed, Global attractivity and periodic nature of a difference equation. World Applied Sciences Journal. 12 (1) (2011), pp 39-47.
[13] H. El-Metwally, Global behavior of an economic model. Chaos, Solitons and Fractals. 33 (2007), pp 994-1005. · Zbl 1196.39008 · doi:10.1016/j.chaos.2006.01.060
[14] H. El-Metwally and M. M. El-Afifi, On the behavior of some extension forms of some population models. Chaos, Solitons and Fractals. 36 (2008), pp 104-114. · Zbl 1152.39307 · doi:10.1016/j.chaos.2006.06.043
[15] H. El-Metwally, E. A. Grove, G. Ladas and McGrath, On the difference equation \(y_{n+1}=\dfrac{y_{n-(2k+1)}+p}{y_{n-(2k+1)}+qy_{n-2l}}\). Proceedings of the 6th ICDE, Taylor and Francis, London. 2004. · Zbl 1065.39012
[16] E. M. Elsayed, Dynamics of recursive sequence of order two. Kyungpook Math. J. 50 (2010), pp 483-497. · Zbl 1298.39004
[17] E. M. Elsayed, A solution form of a class of rational difference equations. Inter. J. Nonlinear Sci. 8 (4) (2009), pp 402-411. · Zbl 1394.39003
[18] E. M. Elsayed, Dynamics of a recursive sequence of higher order. Comm. Appl. Nonlinear Analysis. 16 (2) (2009), pp 37-50. · Zbl 1176.39008
[19] E. M. Elsayed, Behavior of a rational recursive sequences. Studia Univ. ” Babes – Bolyai ”, Mathematica. LVI (1) (2011), pp 27-42. · Zbl 1240.39021
[20] E. M. Elsayed, Solution of a recursive sequence of order ten. General Mathematics. 19 (1) (2011), pp 145-162. · Zbl 1224.39013
[21] E. M. Elsayed, Solution and behavior of a rational difference equations, Acta Universitatis Apulensis . 23 (2010), pp 233-249. · Zbl 1265.39013
[22] E. M. Elsayed, On the Global attractivity and the solution of recursive sequence. Studia Scientiarum Mathematicarum Hungarica. 47 (3) (2010), pp 401-418. · Zbl 1240.39022 · doi:10.1556/SScMath.2009.1139
[23] E. M. Elsayed, Solution and attractivity for a rational recursive sequence. Discrete Dynamics in Nature and Society. Volume 2011 , Article ID 982309, 17 pages. · Zbl 1252.39008
[24] E. M. Elsayed, On the solution of some difference equations. European J. Pure Appl. Math. , · Zbl 1260.39012
[25] A. E. Hamza, S. G. Barbary, Attractivity of the recursive sequence \(x_{n+1}=(\alpha -\beta x_{n})F(x_{n-1},...,x_{n-k}).\;\) Math. Comp. Model. 48 (11-12) (2008), pp 1744-1749. · Zbl 1187.39023 · doi:10.1016/j.mcm.2008.04.011
[26] V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Kluwer Academic Publishers, Dordrecht. 1993. · Zbl 0787.39001
[27] M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures. Chapman & Hall / CRC Press. 2001. · Zbl 0981.39011
[28] A. Rafiq, Convergence of an iterative scheme due to Agarwal et al. Rostock. Math. Kolloq. 61 (2006), pp 95-105. · Zbl 1140.47326
[29] M. Saleh and M. Aloqeili, On the difference equation \(y_{n+1}=A+ \dfrac{y_{n}}{y_{n-k}}\) with \(A<0\). Appl. Math. Comp. 176 (1) (2006), pp 359-363. · Zbl 1096.39011 · doi:10.1016/j.amc.2005.09.023
[30] M. Saleh and M. Aloqeili, On the difference equation \(x_{n+1}=A+ \dfrac{x_{n}}{x_{n-k}}\). Appl. Math. Comp. 171 (2005), pp 862-869. · Zbl 1092.39019 · doi:10.1016/j.amc.2005.01.094
[31] M. Saleh and S. Abu-Baha, Dynamics of a higher order rational difference equation. Appl. Math. Comp. 181 (2006), pp 84-102. · Zbl 1158.39301 · doi:10.1016/j.amc.2006.01.012
[32] D. Simsek, C. Cinar and I. Yalcinkaya, On the recursive sequence \(x_{n+1}=\dfrac{x_{n-3}}{1+x_{n-1}}.\) Int. J. Contemp. Math. Sci. 1 (10) (2006), pp 475-480. · Zbl 1157.39311
[33] C. Wang, S. Wang, L. Li and Q. Shi, Asymptotic behavior of equilibrium point for a class of nonlinear difference equation, Advances in Difference Equations, Volume 2009 , Article ID 214309. 8 pages. · Zbl 1216.39018 · doi:10.1155/2009/214309
[34] C. Wang, S. Wang, Z. Wang, H. Gong and R. Wang, Asymptotic stability for a class of nonlinear difference equation. Discrete Dynamics in Natural and Society . Volume 2010 , Article ID 791610, 10 pages. · Zbl 1192.39015 · doi:10.1155/2010/791610
[35] C. Wang, Q. Shi and S. Wang. Asymptotic behavior of equilibrium point for a family of rational difference equation. Advances in Difference Equations. Volume 2010 , Article ID 505906. 10 pages. · Zbl 1204.39011 · doi:10.1155/2010/505906
[36] I. Yalçınkaya and C. Cinar, On the dynamics of the difference equation \(x_{n+1}=\dfrac{ax_{n-k}}{b+cx_{n}^{p}}\). Fasciculi Mathematici. 42 (2009), pp 133-139.
[37] I. Yalçınkaya, C. Cinar and M. Atalay, On the solutions of systems of difference equations. Advances in Difference Equations. Vol. 2008 , Article ID 143943, doi: 10.1155/2008/ 143943, 9 pages. · Zbl 1146.39023 · doi:10.1155/2008/143943
[38] I. Yalçınkaya, On the global asymptotic stability of a second-order system of difference equations. Discrete Dynamics in Nature and Society. Vol. 2008 , Article ID 860152, doi: 10.1155/2008/ 860152, 12 pages. · Zbl 1161.39013 · doi:10.1155/2008/860152
[39] I. Yalçınkaya, On the difference equation \(x_{n+1}=\alpha +\dfrac{x_{n-m}}{x_{n}^{k}}\). Discrete Dynamics in Nature and Society. Vol. 2008 , Article ID 805460, doi: 10.1155/2008/ 805460, 8 pages. · Zbl 1160.39313 · doi:10.1155/2008/805460
[40] E. M. E. Zayed, Dynamics of the nonlinear rational difference equation \(x_{n+1}=Ax_{n}+Bx_{n-k}+\dfrac{px_{n}+x_{n-k}}{q+x_{n-k}}.\) Europ. J. Pure A ppl. Math. 3 (2) (2010), pp 254-268. · Zbl 1213.39014
[41] E. M. E. Zayed and M. A. El-Moneam, On the rational recursive sequence \(x_{n+1}=ax_{n}-\dfrac{bx_{n}}{cx_{n}-dx_{n-k}}\). Comm. Appl. Nonlinear Analysis. 15 (2008), pp 47-57. · Zbl 1149.39011
[42] E. M. E. Zayed and M. A. El-Moneam, On the rational recursive sequence \(x_{n+1}=\dfrac{\alpha +\beta x_{n}+\gamma x_{n-1}}{ A+Bx_{n}+Cx_{n-1}}.\) Comm. Appl. Nonlinear Analysis. 12 (4) (2005), pp 15-28. · Zbl 1083.39014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.